- Gebundene Ausgabe: 308 Seiten
- Verlag: Palgrave Macmillan; Auflage: 2014 (9. April 2014)
- Sprache: Englisch
- ISBN-10: 1137400722
- ISBN-13: 978-1137400727
- Größe und/oder Gewicht: 14,3 x 2,6 x 22,4 cm
- Durchschnittliche Kundenbewertung: Schreiben Sie die erste Bewertung
- Amazon Bestseller-Rang: Nr. 1.881.937 in Fremdsprachige Bücher (Siehe Top 100 in Fremdsprachige Bücher)
- Komplettes Inhaltsverzeichnis ansehen
An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure (Englisch) Gebundene Ausgabe – 9. April 2014
Es wird kein Kindle Gerät benötigt. Laden Sie eine der kostenlosen Kindle Apps herunter und beginnen Sie, Kindle-Bücher auf Ihrem Smartphone, Tablet und Computer zu lesen.
Geben Sie Ihre Mobiltelefonnummer ein, um die kostenfreie App zu beziehen.
Wenn Sie dieses Produkt verkaufen, möchten Sie über Seller Support Updates vorschlagen?
“An Aristotelian Realist Philosophy of Mathematics is an interesting and challenging book. … Franklin goes through a wealth of material in philosophy, mathematics, and science to motivate and justify his position. … there will still be something for all readers to enjoy due to its large scope and variety of interesting and detailed examples.” (Alex Koo, The Mathematical Intelligencer, Vol. 38, September, 2016)
“The book is comprised of fifteen chapters plus a brief epilogue. … An Aristotelian Realist Philosophy of Mathematics contains many interesting and important discussions, in particular, the discussion of the use of inductive reasoning in mathematics … . This book is provocative and interesting reading for anyone interested in how mathematical entities are related to the physical world and how we gain knowledge of such entities.” (James Davies, Metascience, Vol. 24, 2015)
"A short review cannot do justice to the variety of problems considered and the interesting angles of attack offered by [Franklin's] realist point of view."
-David Guaspari, The New Criterion
"Most of the traditional problems in the philosophy of mathematics arise, in James Franklin's words, out of the 'oscillation between Platonism and nominalism, as if those were the only alternatives' (p. 11). In An Aristotelian Realist Philosophy of Mathematics Franklin develops a tantalizing alternative to these approaches by arguing that at least some mathematical universals exist in the physical realm and are knowable through ordinary methods of access to physical reality. By offering a third option that lies between these extreme all-or-nothing approaches and by rejecting the 'dichotomy of objects into abstract and concrete', Franklin provides potential solutions to many of these traditional problems and opens up a whole new terrain for debate in the philosophy of mathematics (p. 15). The acknowledgement of this by no means new but oft neglected Aristotelian position sheds refreshing new light on debates that have become somewhat stagnant in recent times. Furthermore, by drawing attention to the possibility of an Aristotelian alternative, Franklin opens the way for a whole host of new debates to emerge regarding the correct Aristotelian approach. The scope of the book is ambitious and the overall position defended is controversial in a number of ways. As such, it gives rise to as many new questions as it provides answers. However, this should be seen as a positive, since the many questions that arise are deeply significant and have been neglected by philosophers of mathematics for far too long."
-Max Jones, Philosophica Mathematica
"This book should be part of the personal library of any university-level student of philosophy."
-Jude P. Dougherty, Review of Metaphysics
Über den Autor und weitere Mitwirkende
James Franklin is Professor of Mathematics at the University of New South Wales, Australia and founder of the 'Sydney School' in the philosophy of mathematics. His books include The Science of Conjecture: Evidence and Probability Before Pascal; Corrupting the Youth: A History of Philosophy in Australia; and What Science Knows.
Die hilfreichsten Kundenrezensionen auf Amazon.com
Franklin aims to give an account of mathematics as the science of quantity and of structure. Franklin gives particularly clear definitions of both quantity and structure--something often lacking among contemporary structuralists in my opinion--and this in itself is a very important advance. According to his account, mathematics studies structural universals and quantities. These universals and quantities are the type of thing that can be found in the real world and can be literally had by concrete objects. Of course, not all mathematical structures are had by some concrete object, but it is essential to his account that they could be, i.e. that they are metaphysically possible.
While quantity seems to me to play a less central part in his project, his clear account of structure allows him to take his views a long way. Franklin understands a property to be purely structural just in case it can be defined completely in terms of 'part', 'whole', 'same', 'different', and purely logical vocabulary. The relations of 'part' and 'whole' will probably come into play in geometry, as well as set theory, graph theory, topology, analysis, etc. So, for instance, on this definition, the property of being a Euclidean space could probably be defined purely structurally; see for instance Hilbert's axioms. Also, the Peano axioms seem to describe purely structural relations, since they only invoke logical vocabulary and identity (other than the names for the relations being defined, of course). Franklin gives many more examples, so I refer the reader to his book for a treatment of further cases.
Franklin contrasts his approach with Platonism and nominalism in contemporary philosophy of mathematics. Unlike Platonism, the universals studied by mathematics can be literally instantiated by concrete things in the real world. What mathematics does is study these possibly instantiated structures. Mathematics does not study abstract, particular individuals. Number systems, for instance, would not be cashed out as consisting of abstract individuals (numbers), but as either systems of quantities or as structures which can be instantiated by concrete things. (Franklin's account of number, in fact, cashes out numbers as being relations which are literally instantiated in the world by material heaps and 'unit-making' universals.)
Against nominalism on the other hand, Franklin assumes that there are, in fact, mathematical universals that can be literally shared by different things. Again, Franklin also assumes that there are, in addition to those universals instantiated in the real world, universals which are not instantiated but are at least possibly instantiated.
By his choice of example he shows how contemporary philosophers of mathematics often miss the most central cases of mathematics. Contemporary philosophy of mathematics often has a Platonist bias, focusing on those cases that are less essential for use in real world applications (such as huge sets, large infinities, etc.). This is to the detriment of the most central and basic cases, which are the simple, often discrete and finite structures widely used in real-world applied sciences, and which are less amenable to Platonist interpretation.
He gives a far more plausible account of mathematical knowledge and empirical mathematical application than that offered by most Platonists. He also argues that contemporary philosophy of mathematics tends to not pay enough to attention to how mathematics is actually done, and therefore misses those aspects of mathematical practice that make more sense on Aristotelian view. He shows a much closer parallel between actual mathematical practice and actual empirical scientific practice than is often recognized (for instance, by the unquestionable use of induction, plausible reasoning, and explanation in mathematics; he rightly notes that (in)formal proof is often only the last step in the equation). Franklin goes on to apply the Aristotelian conception of mathematics to many other philosophical issues, such as mathematical necessity, infinity, approximation, and ontology.
With that said, there are several parts of the theory that could be potentially problematic and call for more investigation. Just to shotgun a few of them out:
The reliance on a classical mereology of heaps and arbitrary sums (this is important for his definitions of whole numbers and sets).
The reliance on (immanent) universals, problematic from a trope nominalist perspective such as my own, and which might use a bit more explanation.
The commitment to uninstantiated universals (an idea classically denied by most Aristotelians, including Aristotle himself, and one which moves Franklin's account toward a "semi-Platonism" as he calls it).
His commitment to all mathematical structures being metaphysically possible (this is interesting to me; I bet Franklin's account could be seamlessly extended given a proper account of impossibility, impossible objects, impossible universals, and impossible worlds, and I bet this isn't essential to his view).
Giving a general, unified semantics for mathematical language (it's less than clear from the book how this is to be done; for instance, with the complex and negative numbers, Franklin gives what appear to be examples, or maybe geometrical/economical interpretations. But what would he say are the straight up truth-conditions for, say, -2 + 3i = 2(-1 + 3/2i)? Or (-2)(-3) = 6? Or of more general laws governing number systems?).
Showing more precisely and in individual cases how a more wide range of mathematical concepts are definable either purely structurally or quantitatively (ideally, it'd be nice if we could get to the point of giving a general paraphrase scheme or a general procedure--Franklin's account of set theory being purely structural is suggestive, so maybe we could show how any set-theoretical entity or relation could be defined structurally, and thereby show all mathematics to be interpretable structurally; either this or the last question I hope to work on for my term paper this semester).
The apparently ad hoc fictionalist account of zero and the empty set combined with a realist account of everything else (I can see fictionalists asking why we need the realistic ontology in some cases but not others?)
Related to this last point, some unclarity/implausibility in the theory of ontology and ontological commitment at play, as well as some unclarity about the ontological status of mathematics (if it were made more clear when or why we are committed to some things but not others, and in what way, it'd probably be easier to answer questions such as the last one).
I don't have enough time to spell all these worries out, though if anyone is curious I can explain what I mean, and maybe after reading the book some of these worries will be clear. And I don't think these are damning or insuperable criticisms either; I think they are problems to be investigated, but Franklin's account seems to me to be certainly on the right track.
One last potential criticism that I feel kind of bad about making: I feel like the book doesn't really engage much with what's been said in contemporary neo-Aristotelian metaphysics and ontology. I feel bad about saying that because of the huge swaths of literature the book does, in fact, engage with (the number of works referenced is amazing; one wonders how somebody can read so much). But in certain respects (the mereology for instance, or the role of states of affairs), it seems like the book draws on some concepts with which many current Aristotelians might take issue. And like I said, the book's understanding of ontological commitment could have been a bit more clear; here, engagement with contemporary Aristotelian metaphysics (among others) might have been helpful as well.
Overall though this is an excellent book, and maybe even a game-changer, at least for me. It contains many more interesting ideas and arguments to grapple with than I've been able to discussed here. Whether one buys into it or not, Franklin admirably demonstrates the fruits of an Aristotelian approach, at least on one understanding of that term. He makes use of a wide variety of examples, from a wide variety of real world sciences (including, but very much not limited to, pure mathematics). By doing this he demonstrates how important it is to pay attention to actual empirical results and practice when doing any sort of metaphysical or epistemological investigation into the philosophical status of mathematics. And this seems to me to be one of the most important marks of the general Aristotelian attitude.
Ähnliche Artikel finden